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In this thesis, interaction of an ultrashort single-cycle pulse (USCP) with a bound electron without ionization is studied for the first time. For a more realistic mathematical description of USCPs, Hermitian polynomials and combination of Laguerre functions are used for two different single-cycle excitation cases. These single-cycle pulse models are used as driving functions for the classical approach to model the interaction of a bound electron with an applied field. Two different new novel time domain modification techniques are developed for modifying the classical Lorentz damped oscillator model in order to make it compatible with the USCP excitation. In the first technique, a time dependent modifier function (MF) approach has been developed that turns the Lorentz oscillator model equation into a Hill-like equation with non-periodic time varying damping and spring coefficients. In the second technique, a time dependent convolutional modifier function (CMF) approach has been developed for a close resonance excitation case. This technique provides a continuous updating of the bound electron motion under USCP excitation with CMF time upgrading of the oscillation motion for the bound electron. We apply each technique with our two different driving model excitations. Each model provides a quite different time response of the bound electron for the same applied time domain technique. Different polarization response will subsequently result in relative differences in the time dependent index of refraction. We show that the differences in the two types of input oscillation fields cause subduration time regions where the perturbation on the real and imaginary part of the index of refraction dominate successively.